interactive visualization of 2d persistence modules · interactive visualization of 2d persistence...

Interactive Visualization of 2D Persistence Modules

Michael Lesnick, Columbia

w/ Matthew Wright, St. Olaf

IAS, November 7, 2015

Persistent homology:

• provides invariants of data called barcodes

• used for exploratory data analysis/visualization

• many practical tools are available

Fig. by Ulrich Bauer.

Multi-D Persistent Homology

• Associates to data a multi-parameter family of topology spaces.

• arises naturally in applications

• no practical tools yet available

Fig. by Matthew Wright.

RIVET: A practical tool for interactive visualization of 2D persistenthomology.

• expected public release: winter 2016

• paper this month

Mathematical contributions:

• Theoretical/algorithmic framework for efficient queries of barcodesof 1-D slices of 2-D persistence objects.

• O(n3) algorithm for computation of bigraded Betti numbers.

• Algorithms for computing 1-parameter families of barcodes.

Agenda:

• Introduce multidimensional persistent homology

• Explain our tool

• Briefly discuss theoretical and algorithmic underpinnings

1-D Persistent Homology

Persistent HomologyPersistent homology associates barcodes to data.

Data:• Finite metric space (point cloud data)• function γ : T → R, T an arbitrary topological space.

Persistence Diagrams

Usually all intervals in a barcode of the form [b, d).

Then we can regard the barcode as a collection of points (b, d) in theplane with b < d.

constructing barcodes

Pipeline for 1-D Persistence

Filtrations and Persistence Modules

A filtration F is a collection of topological spaces {Fa} indexed by R(or by Z) such that Fa ⊆ Fb whenever a ≤ b.

In Z-indexed case, this is a diagram of spaces:

· · · ↪→ F0 ↪→ F1 ↪→ F2 ↪→ · · ·

Fix a field k.

A persistence module M is a collection of k-vector spaces {Ma}indexed by R (or by Z) and commuting linear maps

{M(a, b) : Ma →Mb}a<b.

· · · →M0 →M1 →M2 → · · ·

Rips Filtrations

For P a metric space, and a ∈ R, define simplicial complex Rips(P )a by:

• Vertex set of Rips(P )a is P .

• Rips(P )a contains edge [q, r] iff dP (q, r) ≤ a2 .

• Rips(P )a is the clique complex on this 1-skeleton.

Rips(P )a ⊆ Rips(P )b whenever a ≤ b, so we obtain a filtration

Rips(P ) = {Rips(P )a}a∈R.

[Fig. from M. Wright’s “Introduction to Persistent Homology,” Youtube.]


Applying ith homology to each space and inclusion map in a filtrationyields a persistence module.

structure theorem for persistent homology (Z-indexed case)

For a < b ∈ Z,

• call [a, b) a discrete interval,

• define the interval module I [a,b) by

· · · // 0 // kIdk // k

Idk // · · · Idk // k // 0 // 0 // · · ·

a b

• define infinite discrete intervals, interval modules similarly.

Decomposition Thm. [Webb ’85]: For M a Z-indexed persistencemodule w/ finite dim. vector spaces, ∃ unique collection of discreteintervals B(M) s.t.

M ' ⊕J∈B(M)IJ

We call B(M) the barcode of M .

Persistent Homology

Figure by Ulrich Bauer.

Stability of PH of PCD

Persistent Homology of PCD is stable with respect to Gromov-Hausdorffdistance on finite metric spaces.

Limits of Stability

Persistent homology is NOT stable with respect to outliers.

This leads us to multi-D persistence.

Multi-D Persistent Homology

Pipeline for 2D Persistence

Bifiltrations

• Define a partial order on R2 by

(a1, a2) ≤ (b1, b2) iff ai ≤ bi for i = 1, 2;

• A bifiltration is a collection of topological spaces {Fa} indexed byR2 (or by Z2) such that Fa ⊆ Fb whenever a ≤ b.

A 2-D persistence module M is a collection of k-vector spaces {Ma}indexed by R2 (or by Z2) and commuting linear maps

{M(a, b) : Ma →Mb}a<b.

......

...

· · · //M1,3//

OO

M2,3//

OO

M3,3//

OO

· · ·

· · · //M1,2//

OO

M2,2//

OO

M3,2//

OO

· · ·

· · · //M1,1//

OO

M2,1//

OO

M3,1//

OO

· · ·

...

OO

...

OO

...

OO


Applying ith homology to a bifiltration F yields a 2D persistence moduleHiF .

Point cloud data → Bifiltration

Limits of Stability

Persistent homology is NOT stable with respect to outliers.

This leads us to multi-D persistence.

Point cloud data → BifiltrationFor P a finite metric space, let γ : P → R be a codensity function onP .• i.e., γ is high at outliers and low at dense points.

• example: fix K > 0 and let γ(x) = distance to the Kth nearestneighbor of P .

For a ∈ R, define the a-sublevelset

γa := {y ∈ P | γ(y) ≤ a}.

For (a, b) ∈ R2, letF(a,b) = Rips(γa)b.

{F(a,b)}(a,b)∈R2 , together w/ inclusion maps, is a bifiltration.

Point cloud data → BifiltrationFor P a finite metric space, let γ : P → R be a codensity function onP .• i.e., γ is high at outliers and low at dense points.• example: fix K > 0 and let γ(x) = distance to the Kth nearest

neighbor of P .

For a ∈ R, define the a-sublevelset

γa := {y ∈ P | γ(y) ≤ a}.

For (a, b) ∈ R2, letF(a,b) = Rips(γa)b.

{F(a,b)}(a,b)∈R2 , together w/ inclusion maps, is a bifiltration.

Barcodes of Bifiltration?

Can we define the barcode of 2D persistence module as a collection ofnice regions in R2?

Not without making some significant compromises.

Theorem [Krull-Schmidt]: For M a finitely presented 2D persistencemodule, ∃ collection of indecomposables M1, . . .Mk, unique up to iso.,such that:

M ' ⊕ki=1Mi.

Lesson from quiver theory: The set of possible Mi is extremelycomplicated.

· // · // · // · // · // ·

·

OO

·

OO

The upshot: There’s no entirely satisfactory way to define barcode of M .

Potentially useful invariants of 2-D persistence modules

Our tool visualizes three invariants of a 2D persistence module:

• Dimension of vector space at each index

• Barcodes of 1-D affine slices of the module

• Multigraded Betti numbers

Dimension of vector space at each index:

• simple, intuitive, easy to visualize,

• Can compute in time cubic in the size of the input,

• tells us nothing about persistent features,

• not stable.

Barcodes of 1-D Slices

• Let L be an affine line in R2 w/ non-negative slope.

• Restriction of M to L is a 1-D persistence module ML.

• Thus ML has a barcode B(ML), a set of intervals in L.

Barcodes B(ML) is stable [Landi 2014, Cerri et al. 2011, Cerri et al.2013].

Multigraded Betti NumbersFor M an n-D persistence module, i ∈ {0, 1, . . . n}, one defines functionsξi(M) : Rn → N by

ξi(M)a = dim Tori(M,k)a.

For a ∈ Rn,• ξ0(M)(a) is the dimension of what is born at a,• ξ1(M)(a) is the dimension of what dies at a,

Bigraded Betti NumbersLet M be a Z2-indexed persistence module, (a, b) ∈ Z2.

M(a−1,b) //M(a,b)

M(a−1,b−1) //

OO

M(a,b−1)

OO

We have induced maps

M(a−1,b−1)split−−→M(a−1,b) ⊕M(a,b−1)

merge−−−→M(a,b)

withmerge ◦ split = 0.

For i = 0, 1, 2, define ξi(M) : R2 → N by

ξ0(M)(a, b) = dimM(a,b)/ im merge

ξ1(M)(a, b) = dim ker merge / im split

ξ2(M)(a, b) = dim ker split .

An Example• Codensity-Rips Bifiltration on noisy PCD circle• 240 points• ∼200,000 simplices• Round distances and codensities to lie on a 30× 30 grid,• 1st persistent homology

Mathematical contributions:

• Theoretical and algorithmic framework for interactive visualizationof barcodes of 1-D slices,

• Novel algorithm for fast computation of multigraded Betti numbers.

• Algorithms for computing 1-parameter families of barcodes.

Data Structure for Interactive Visualization

Let M be a 2D persistence module.

We define a data structure A(M), the augmented arrangement of M ,on which can perform fast queries of B(ML) for any line L.

A(M) consists of:

• a line arrangement in (0,∞)× R,

• for each 2-cell e, a collection Pe of pairs (a, b) ∈ R2 × (R2 ∪∞).

We call the Pe the barcode template at e.

Point-Line DualityLet Λ = the set of affine lines with finite, positive slope.

Define dual maps

D` : Λ→ (0,∞)×R, Dp : (0,∞)×R→ Λ

D`(y = mx + b) = (m,−b)Dp(m, b) = (y = mx− b).

L′

u

v

w

Dp(u)

Dp(v)Dp(w)

D`(L′)

Push Maps

For any L ∈ Λ, we have a map

pushL : R2 → L,

which sends each u ∈ R2 to the closest point of L above or to the rightof u.

L

upushL(u) v

pushL(v)

Main Theorem

Theorem [L., Wright 2015]: For M a 2-D persistence module, L ∈ Λ,and e any 2-D coface of the cell in A(M) containing D`(L),

B(ML) = {[ pushL(a), pushL(b) ) | (a, b) ∈ Pe},

L

I1

I2

a1 b1

a2

b2

Ex: B(ML) = {I1, I2} Pe = {(a1, b1), (a2, b2)}

complexity

Queries

Let κ be minimal number of vertices in a rectangular grid containingsupp ξ0(M) ∪ supp ξ1(M).

Proposition: For a generic line L, we can perform a query of A(M) intime O(log κ+ |B(ML)|).

Constructing the Augmented Arrangement

Proposition: For fixed i and F a bifiltration of size m, constructingA(HiF) requires

O(m3κ+ (m+ log κ)κ2)

elementary operations and

O(m2 +mκ2)

storage.

Remarks:

• most expensive steps are embarrassingly parallizable

• Algorithm involves computation of the Betti numbers.

Preliminary Timing ResultsA snapshot from our current code. Several important optimizations arenot yet implemented; substantial speedups are ahead.

Data:

• Codensity-Rips Bifiltration on noisy PCD circle

• codensity and distance each coarsened to lie on 20x20 grid

• Truncated Rips filtration on 400 points,

• 6,00,000 simplices

• (slow) 800 MHz processor

Betti numbers:

• H0: 4 Sec.

• H1: 13 minutes

Augmented arrangement

• H0: 11 minutes

• H1: 16.4 hours

thank you!!

interactive visualization of 2d persistence modules · interactive visualization of 2d persistence...

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